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Properties

Label 4-3024e2-1.1-c1e2-0-21
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 4·19-s + 7·25-s + 12·29-s − 20·31-s − 22·37-s + 18·47-s + 9·49-s − 24·53-s + 6·59-s + 30·83-s − 8·103-s + 10·109-s − 36·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.917·19-s + 7/5·25-s + 2.22·29-s − 3.59·31-s − 3.61·37-s + 2.62·47-s + 9/7·49-s − 3.29·53-s + 0.781·59-s + 3.29·83-s − 0.788·103-s + 0.957·109-s − 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.462813678\)
\(L(\frac12)\) \(\approx\) \(2.462813678\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.17.a_ah
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.31.u_gg
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.37.w_hn
41$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \) 2.41.a_adb
43$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.43.a_ach
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.47.as_gt
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.61.a_aeg
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.71.a_by
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.73.a_adu
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.a_al
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.83.abe_pb
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.97.a_ec
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.872622283220259127185845127535, −8.684465308900046124477111975093, −8.131851315225435693770630618465, −7.72338518505025587316238289617, −7.62112508302919210895397950862, −6.92112389974738081050197346690, −6.61181337334973260989875117741, −6.58768498218918007471176386339, −5.57312619810038405231423009757, −5.42207609828340050113608614520, −5.15556415856095759135825336559, −4.65362491375754674485683089362, −4.32635082558230335095869552433, −3.78689889525344726823307404797, −3.34211146395643793259553752300, −2.88612970688943832530762588226, −2.05829192424773217696333797947, −1.88721971139079786726223374655, −1.34982146150484582549255033383, −0.49525080591955792677957552443, 0.49525080591955792677957552443, 1.34982146150484582549255033383, 1.88721971139079786726223374655, 2.05829192424773217696333797947, 2.88612970688943832530762588226, 3.34211146395643793259553752300, 3.78689889525344726823307404797, 4.32635082558230335095869552433, 4.65362491375754674485683089362, 5.15556415856095759135825336559, 5.42207609828340050113608614520, 5.57312619810038405231423009757, 6.58768498218918007471176386339, 6.61181337334973260989875117741, 6.92112389974738081050197346690, 7.62112508302919210895397950862, 7.72338518505025587316238289617, 8.131851315225435693770630618465, 8.684465308900046124477111975093, 8.872622283220259127185845127535

Graph of the $Z$-function along the critical line